I. INTRODUCTION In a previous paper [1], we establish a general method to calculate eigenvalues of Casimir operators of any order. Although it is work

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1 NON-RECURSIVE MULTIPLICITY FORMULAS FOR A N LIE ALGEBRAS H. R. Karadayi Dept.Physics, Fac. Science, Tech.Univ.Istanbul 8066, Maslak, Istanbul, Turkey Internet: karadayi@faraday.z.fe.itu.edu.tr Abstract It is shown that there are innitely many formulas to calculate multiplicities of weights participating in irreducible representations of A N Lie algebras. On contrary to the recursive character of Kostant and Freudenthal multiplicity formulas, they provide us systems of linear algebraic equations with N-dependent polinomial coecients. These polinomial coecients are in fact related with polinomials which represent eigenvalues of Casimir operators.

2 I. INTRODUCTION In a previous paper [1], we establish a general method to calculate eigenvalues of Casimir operators of any order. Although it is worked for A N Lie algebras in ref.(1), the method works well also for other Classical and Exceptional Lie algebras because any one of them has a subgroup of type A N. Let I(s) be a Casimir operator of degree s and s 1 + s + ::: + s k s (I:1) be the partitions of s into positive integers on condition that s 1 s ::: s k : (I:) The number of all these partitions of s is given by the partition function p(s) which isknown to be dened by 1Y 1 1 (1 x n ) p(s) x s : n=1 We also dene (s) to be the number of partitions of s into positive integers except 1. We know then that we have at least a (s) number of ways to represent the eigenvalues of I(s) in the form of polinomials s=0 P s1;s;::sk ( + ;N) (I:3) : where + is the dominant weight which species the representation R( + ) for which we calculate eigenvalues. In ref.(1), we give, for orders s=4,5,6,7 all the polinomials (I.3) explicitly and show that they are in coincidence with the ones calculated directly as traces of the most general Casimir operators []. This, on the other hand, brings out another problem of computing multiplicities of all other weights participating within the representation R( + ). This problem has been solved some forty years ago by Kostant [3] and also Freudenthal [4]. In spite of the fact that they are quite explicit,the multiplicity formulas of Kostant and Freudenthal expose serious diculties in practical calculations due especially to their recursive characters. It is therefore worthwhile to try for obtaining some other non-recursive formulas to calculate weight multiplicities. Let us recall here that,since the last twenty years [5], there are eorts which give us weight multiplicities in the form of tables, one by one. We will give in this work general formulas with the property that they depend only on multiplicities and rank N. This will give us the possibility to obtain, by induction on values N=1,,..., as many equations as we need to solve all unknown parameters, i.e. multiplicities. II. WEYL ORBITS AND IRREDUCIBLE REPRESENTATIONS We refer the book of Humphreys [6] for Lie algebraic technology which we need in the sequel. The indices like i 1 ;i ;:: take values from the set I 1; ; ::N while I 1 ;I ;:: from S 1; ; ::N; N +1. The essential gures are simple roots i, fundamental dominant weights i and the fundamental weights dened by 1 1 I I 1 I 1 ; I =;3; ::N +1 : (II:1) Any dominant weight then has the decompositions + = N i=1 r i i ; r i Z + (II:) where Z + is the set of positive integers including zero. (II.) can be expressed equivalently in the form + = i=1 q i i ; =1;; ::N (II:3)

3 3 together with the conditions q 1 q ::: q > 0 : The weight space decomposition of an irreducible representation R( + ) has now the form R( + ) = ( + ) + + Sub(+ + ) m( + < + ) ( + ) (II:4) (II:5) where m( + < + ) is the multiplicity ofweight + within the representation R( + ) and ( + ) represents its corresponding Weyl orbit. Sub( + ) here is the set of all sub-dominant weights of +. Although the concept of Casimir eigenvalue is known to be dened for representations of Lie algebras, an extension for Weyl orbits has been made, in ref.(1), with the aid of denition ch s () () s : (II:6) This allows us to make the decompositions ch s () = s1;s;::sk (s 1 ) (s ):::(s k ) cof s1;s;::sk (II:7) in terms of generators (s) dened by (s) N+1 I=1 ( I ) s : (II:8) We consider here the partitions (I.1) for any order s. It is then known that the coecients in (II.7) are expressed in the form cof s1;s;::sk cof s1;s;::sk ( + ;N) ; i.e. as N-dependent polinomials for a Weyl orbit ( + )ofa N Lie algebras. One further step is to propose the existence of some polinomials P s1;s;::sk ( + ;N) satisfying following equations: P s1 s::sk ( + ;N) cof s1s::sk( + ;N) cof s1s::sk ( k ;N) dimr( k ;N) dimr( + ;N) P s1s::sk( k ;N) : (II:9) Note here that and also dimr( k ;N)= cof s1s::sk ( i ;N) 0 ; i<k (II:10) (N+ 1)! k! (N+1 k)! ; k =1;; ::N: (II:11) where dimr( + ) is the number of weights within the representation R( + ), i.e. its dimension. By examining (II.9) for a few simple representations, a (s) number of polinomials can be obtained for each particular value of order s. The ones for s=4,5,6,7 are given in ref.(1) explicitly. III. THE MULTIPLICITY FORMULAS Our strategy here is to use the equations (II.9) in the form of s1;s;::s k ( + ;N) P s1s::sk ( + ;N) dimr( + ;N) cof s1s::sk ( k ;N) P s1s::sk ( k ;N) dimr( k ;N) cof s1s::sk ( + ;N) with which we obtain, for each particular partition (I.1) of degree s, a multiplicity formula s1;s;::s k ( + ;N) 0 (III:1) (III:)

4 4 for weight multiplicities within the representation R( + ) and for any values of rank N >. We think + here as in the form of (II.3). In addition to the ones given in (II.10), the following expressions can be borrowed from ref.(1): These are sucient to give cof s ( 1 ;n)=1 ; cof s1;s ( ;n)= (s 1+s )! s 1! s! ; s 1 >s ; cof s1;s1 ( ;n)= (s 1+s 1 )!! s 1! s 1! ; cof s1;s;s3 ( 3 ;n)= (s 1+s +s 3 )! s 1! s! s 3! ; s 1 >s >s 3 ; cof s1;s;s ( 3 ;n)= (s 1+s +s )!! s 1! s! s! ; s 1 >s ; cof s1;s1;s1 ( 3 ;n)= (s 1+s +s 1 )! 3! s 1! s 1! s 1! : 7 s=4 (s) =1 (III:3) dierent multiplicity formulas originating from the same form (III.). To proceed further let us take (II.5) in the form R( + ) where we usually dene the height p(h + ) =1 m() ( ) ; Sub(+ + ) (III:4) h + i=1 q i (III:5) : for (II.3) and p(h +) is just the partition function dened above. A further focus here is to make a gradation for elements of the set Sub(s 1 )by assigning, for each one of them, a grade (s 1 ;s ; ::; s k )=1;; ::; p(s) (III:6) as being in line with the conditions (I.). Then, it is clear in (III.4) that m() 0 ; ( ) >( + ) : (III:7) Note also that all dominant weights within a Sub( + )must have the same height. In view of (III.4), one knows both cof s1s::sk ( + ;N) and also dimr( + ;N) as linear superpositions of multiplicities m() with N-dependent polinomial coecients. It is noteworthy here that dimensions of Weyl orbits are already known due to a permutational lemma given in ref.(1). We,hence, give in the following our results for 1 multiplicity formulas extracted from (III.) for s=4,5,6,7: 7 ( + ;N)=cof 7 ( + ;N) g(n)+ dimr( + ;N) ( 70 f 7 7 (N) (7; + ;N) f5(n) 7 (5; + ;N) (; + ;N) f43(n) 7 (4; + ;N) (3; + ;N) f3(n) 7 (3; + ;N) (; + ;N) ) (III:8)

5 5 5 ( + ;N)=cof 5 ( + ;N) N (N +)g(n)+ dimr( + ;N) ( 5040 f 5 7 (N) (7; + ;N) 504 f 5 5 (N) (5; + ;N) (; + ;N) 5040 f 5 43 (N) (4; + ;N) (3; + ;N) + 50 f3(n) 5 (3; + ;N) (; + ;N) +4f 5 5 (N) (5; + ;N) 10 f 5 3 (N) (3; + ;N) (; + ;N)) (III:9) 43 ( + ;N)=cof 43 ( + ;N)1N(N+) g(n)+ dimr( + ;N) ( f 43 7 (N) (7; + ;N) f 43 5 (N) (5; + ;N) (; + ;N) 5040 f (N) (4; + ;N) (3; + ;N) f3(n) 43 (3; + ;N) (; + ;N) 7 f 43 3 (N) (3; + ;N)) (III:10) 3 ( + ;N)=co (+;N)4N(N+) g(n)+ dimr( + ;N) ( (N) (7; + ;N) (N) (5; + ;N) (; + ;N) (N) (4; + ;N) (3; + ;N) (N) (3; + ;N) (; + ;N) (N) (5; + ;N) (N) (3; + ;N) (; + ;N) 7 3 (N) (3; + ;N)) (III:11) 6 ( + ;N)=cof 6 ( + ;N) 5 g 6 (N) + dimr( + ;N)(N+1)g(N)+ dimr( + ;N) ( 3040 f 6 6 (N) (6; + ;N) f 6 4(N) (4; + ;N) (; + ;N) (III:1) f 6 33(N) (3; + ;N) f 6 (N) (; + ;N) 3 ) 4 ( + ;N)=co ( + ;N) 67 N (N +1)(N+) g 6 (N)+ dimr( + ;N) N (N +1)(N+)(7 N +14 N + 47) g 6 (N) + dimr( + ;N) ( (N) (6; + ;N) (N) (4; + ;N) (; + ;N) (N) (3; + ;N) (N) (; + ;N) (N) (4; + ;N) 5040 (N) (; + ;N) 84 (N) (; + ;N)) (III:13)

6 6 33 ( + ;N)=cof 33 ( + ;N) 16 N (N +1)(N+) g 6 (N) dimr( + ;N)5N(N+1)(N+) g 6 (N)+ dimr( + ;N) ( 1510 f 33 6 (N) (6; + ;N) 6800 f 33 4 (N) (4; + ;N) (; + ;N) 5040 f (N) (3; + ;N) f 33 (N) (; + ;N) 3 ) (III:14) ( + ;N)=co ( + ;N) 576 N (N +1)(N+)g 6 (N)+ dimr( + ;N) N (N +1) (N+)(5 N +10N+ 3) g 6 (N) + dimr( + ;N) ( (N) (6; + ;N) (N) (4; + ;N) (; + ;N) (N) (3; + ;N) (N) (; + ;N) (N) (4; + ;N) 160 (N) (; + ;N) +36 (N) (; + ;N)) (III:15) 5 ( + ;n ) =cof 5 ( + ;n) g 5 (n) dimr( + ;n) ( 4 f 5 5 (n) (5; + ;n) 10 f 5 3(n) (3; + ;n) (; + ;n)) (III:16) 3 ( + ;n ) =cof 3 ( + ;n)3g 5 (n)+ dimr( + ;n) ( 360 f 3 5 (n) (5; + ;n) 60 f 3 3 (n) (3; + ;n) (; + ;n) +5 f 3 3 (n) (3; + ;n)) (III:17) 4 ( + ;N)= cof 4 ( + ;N) 10 g 4 (N) + dimr( + ;N)(N+1)g 4 (N)+ dimr( + ;N) ( 70 f 4 4 (N) (4; + ;N) (III:18) 70 f 4 (N) (; + ;N) ) ( + ;N)=cof ( + ;N) 40 (N +1) g 4 (N) (N+1) g(n)dimr( + ;N)+ dimr( + ;N) ( 1440f4(N)(4; + ;N) 70f(N)(; + ;N) +10f(N)(; + ;N)) (III:19) where the quantities dened by (s ; + ;N) N+1 ( I ) s can be calculated explicitly via re-denitions I=1 1+r i i i+1 ; i I of the parameters r i in (II.). Note here that (1; + ;N) 0. All coecient polinomials are given in appendix.

7 7 IV. AN EAMPLE : R( ) Now it will be instructive to demonstrate the idea in an explicit example, chosen, say, from the set Sub(9 1 ) with the gradation (III.6) from 1 to p(9)=30 for its 30 elements all having the same height (= 9). In the notation of parameters q i dened in (II.3), (III.4) turns out to be with dimr( )= m(0) 4 + m(1) 36 + m() 70 + m(3) 40 + m(4) 5040 and with a straightforward computation R( )=m(0) (3; ; 1; 1; 1; 1) + m(1) (; ; ; 1; 1; 1) + m() (3; 1; 1; 1; 1; 1; 1) + m(3) (; ; 1; 1; 1; 1; 1) + m(4) (; 1; 1; 1; 1; 1; 1; 1) + m(5) (1; 1; 1; 1; 1; 1; 1; 1; 1) (N 4) (N 3) (N ) (N 1) N (N +1) (N 4) (N 3) (N ) (N 1) N (N +1) (N 5) (N 4) (N 3) (N ) (N 1) N (N +1) (N 5) (N 4) (N 3) (N ) (N 1) N (N +1) (N 6) (N 5) (N 4) (N 3) (N ) (N 1) N (N +1) + m(5) (N 7) (N 6) (N 5) (N 4) (N 3) (N ) (N 1) N (N +1) : (; ;N)= g(;n) ( N N +4 N 3 +N 4 ) (4; ;N)= 1 g(4;n) ( N N N N 4 05 N N 6 +4N 7 +3 N 8 ) 1 (6; ;N)= g(6;n) ( N N N N N N N N N N N 11 +3N 1 ) (3; ;N)= 43 g(3;n) ( N 31 N 4 N 3 + N 4 ) (5; ;N)= 43 ( N N g(5;n) 556 N N N N 6 16 N 7 +3N 8 ) (7; ;N)= 88 g(7;n) ( N N N N N N N N N N N 11 +9N 1 ) where g(s; N) =3 s (s+1)(n+1) s 1 :

8 8 In this example, we have supressed explicit N-dependences though we recall that all expressions are valid for N>6. It is thus seen that all the formulas given from (III.8) to (III.19) above give rise to the same result m(1) = m() = m(0) 5 m(0) m(3) = 10 m(0) m(4) = 35 m(0) m(5) = 105 m(0) for which one always knows that m(0) = 1. As can be easily investigated by Weyl dimension formula, this also leads us to the result. dimr( )= (N 4) (N 3) (N ) (N 1) N (N +1) (N+)(N+3) V. CONCLUSIONS We obtained here 1 multiplicity formulas for a weight within an irreducible representation of A N Lie algebras. The method is based essentially on some polinomials representing eigenvalues of Casimir operators. These polinomials has been given for degrees s=4,5,6,7 in a previous work. If one further considers the Casimir operators of higher degrees, it is clear that one can obtain, in essence, an innite number of multiplicity formulas. On the other hand, Casimir eigenvalues of any other Classical or Exceptional Lie algebra having a subalgebra of type A N can be obtained by the aid of these polinomials. It could therefore be expected that the multiplicity formulas given above are to be generalized further to these Lie algebras. Another point which seems to be worthwhile to study is the hope that such a framework will prove useful also for Lie algebras beyond the nite ones. To this end, the crucial point will be to x a convenient subalgebra which underlies the innite dimensional one. For instance, one can think that an E 8 multiplicity formula could be reformulated in terms of its subalgebra A 7 or more suitably A 8. But what is more intriguing is to consider the same possibility,say, for hyperbolic Lie algebra E 10. It is clear that to investigate these possibilities shed some light on the multiplicity problems of innite Lie algebras for which quite little is known about their multiplicity formulas in general. REFERENCES [1] Karadayi H.R and Gungormez M: Explicit Construction of Casimir Operators and Eigenvalues:II, physics/mathematical methods in physics/961100, submitted to Jour.Math.Phys. [] Karadayi H.R and Gungormez M: Explicit Construction of Casimir Operators and Eigenvalues:I, hep-th/ ,submitted to Jour.Math.Phys. [3] Kostant B.; Trans.Am.Math.Soc. 93 (1959) [4] Freudenthal H. ; Indag.Math. 16 (1954) and Freudenthal H. ; Indag.Math. 18 (1956) [5] Patera J. and Sanko D. : Tables of Branching Rules for Representations of Simple Lie Algebras, L'Universite de Montreal, 1973 McKay W. and Patera J. : Tables of Dimensions, Indices and Branching Rules for Representations of Simple Algebras, Dekker, NY 1981 Slansky R : Group Theory for Unied Model Building, Physics Reports [6] Humphreys J.E: Introduction to Lie Algebras and Representation Theory, Springer-Verlag (197) N.Y.

9 9 APPENDI We give here N-dependent coecient polinomials encountered in the multiplicity formulas given in section (III). f 7 7 (N) =N 6 +6N 5 +50N N N N + 10 f5(n) 7 =N 5 +5N 4 +1N 3 +43N 70 N 96 f43(n) 7 =N 5 +5N 4 +9N 3 +7N +6N+60 f3(n) 7 =N 4 +8N 3 5N 6 N 15 f 5 7 (N) =N 7 +7N 6 +31N 5 +85N 4 +16N 3 36 N 19 N f 5 5 (N) =N 8 +8N 7 +3N 6 +80N N N N +7N f 5 43 (N) =6N 6 +36N 5 N +13N N N f 5 3(N) =N 7 +7N 6 70 N N N 134 N 840 f 5 3 (N) =N N 9 19 N 8 39 N N N N N N 1680 N f 5 5 (N) =N N 10 N N N N N N N N N f 43 7 (N) =N 7 +7N 6 +19N 5 +5N 4 +76N N + 10 N f 43 5 (N) =6N 6 +36N 5 +13N N 3 N N f (N) =N 8 +8N 7 +16N 6 16 N N N N 8060 N 8400 f 43 3(N) =N 7 +14N N N N N N f 43 3 (N) =(N 5) (N 4) (N 3) (N ) N (N +1) 3 (N+)(N+4) (N+5)(N+6)(N+7) 7 (N) =N 6 +1N 5 +11N 4 36 N 3 67 N 30N 5 (N) =N 7 +7N 6 70 N N N 134 N (N) =N 7 +14N N N N N N (N) =N 8 +8N 7 3N N N N N N (N) =(N 5) (N 4) N (N +1) (N+)(N+6)(N+7)(N +N 1) 3 (N) = (N 4) (N 5) N (N +1)(N+)(N+6)(N+7)(N 4 +4N 3 +6N +4 N+ 5) 3 (N) =(N 5) (N 4) N (N +1)(N+)(N+6)(N+7)(5N 7 +35N 6 14 N 5 40 N N N N + 130) f 6 6 (N) =N 5 +5N 4 +5N 3 +55N +58N+4 f4(n) 6 =N 4 +4N 3 +7N +6N 18 f33(n) 6 =3N 4 +1N 3 +7N 10 N +7 f(n) 6 =N 3 +3N 4N 6

10 10 6 (N) =N 7 +7N 6 +1N 5 +35N 4 +14N 3 4 N 36 N 4 (N) =N 8 +8N 7 +8N 6 +56N N N N N (N) =N 6 +6N 5 +5N 4 0 N 3 0 N +16N+96 (N) =N 7 +14N 6 3N N N N 16 N (N) =N N N 9 04 N N N N N N N N (N) =N 10 +0N 9 +3N N N N 5 + =(N+1) g 6 (N) 737 N N N 70 N f 33 6 (N) =3N 7 +1N 6 +49N 5 +35N 4 +56N N N f 33 4 (N) =N 6 +6N 5 +5N 4 0 N 3 0 N +16N+96 f (N) =N 8 +8N 7 11 N N N N 03 N 3840 f 33 (N) =4N 5 +0N 4 19 N N +78N (N) =N 6 +6N 5 +7 N 4 1 N 3 6 N 1 N 4 (N) =N 7 +14N 6 3N N N N 16 N (N) =4N 5 +0N 4 19 N N +78N+ 180 (N) =N 8 +8N 7 7 N N 5 79 N N N N (N) =N 10 +0N 9 +3N N N N N N N 70 N (N) =N N 10 +7N 9 67 N N N N N N N N (N) =5N N N N N N N N N N N N N N f 5 5 (N) =N 4 +4N 3 +11N +14N+6 f 5 3(N) =N 3 +3N +N 1 f 3 5 (N) =N 3 +3N +N 1 f 3 3 (N) =N 4 +4N 3 +6N +4 N+5 f 3 3 (N) =(N 3) (N ) (N +1) 3 (N+4)(N+5) f 4 4(N)=N 3 +3N +4N+ f 4 (N) =N +4N 1 f 4 (N) =N 3 +6N +3 N 1 f (N) =N 4 +4N 3 8N+13 f (N) =N 7 +7N 6 +8N 5 30 N 4 59 N 3 N +50N+4